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When Penrose, Hawking, and others talk about the beginning of something in a black hole or the big bang, they state that these theories include the idea of the beginning of time. My question is, what is the beginning of time? What is time, a function of human consciousness, a variable in equations involving the speed of light, an undiscovered constant, etc? When we talk of space-time, what could it mean objectively?

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The basic fact that give rise to the perception of time is change. Temporal duration is experienced as a measure of the change of something from one state to the next in the context of other such changes all around it. It is conceptualized with a unit of regularity such as the period of a pendulum swinging, from which other durations can be measured and the concept of time formed in terms of the usual process of omitted measurements.

Like any other concept, time requires both the "external" facts of entities changing, which give rise to it and are referred to by the concept, and our awareness of it in consciousness. It is objective as conscious awareness of that which is -- not subjective fantasy, and not something intrinsic and completely apart from consciousness. Only the facts that give rise to it are independent of consciousness.

Time is therefore "in the universe", not the other way around as "the universe in time", and originates "locally" through local observation, not as an attribute of existence as a whole. Changes in the galaxies, etc. are measured in terms of the concept of time with which you start.

There is no "beginning of time" because there is no beginning of existence. "Existence" refers to everything that is, was, or will be regardless of its state and changes, not the current physical universe. Existence has no beginning in non-existence. Non-existence is an abstraction from existence as you think of something not there, which once was, might have been, etc. There is no "somewhere else" in which non-existence metaphysically resides and from which existence could have arisen.

Time is therefore eternal, meaning that the facts that give rise to it are in principle always available for measurement of duration within an existence that has always been, whatever form it may be in. That does not mean the existence of an "infinity", only that time is an attribute that is always within existence.

Time variables in equations of physics refer to measurements of time in some unit, and presuppose the conceptual understanding of time before they can be related to other physical quantities in mathematically describing their change.

Space-time is a scientific concept, not a matter of general philosophy. It refers to the abstract mathematical "space" of both spatial and time variables relating time and spacial coordinates in the equations of physics. That ranges from the simplest kinematic equations, to Newtonian dynamics and electromagnetic fields, to the most abstract tensor equations of general relativity relating space coordinates to time in ways more complex than simple explicit functions. In essence, the simplest example illustrating this is plotting distance and time on a two-dimensional space on a piece of paper (or computer monitor). But it is much more abstract and complex than that in the realm of advanced mathematical physics which routinely deals with four dimensional "spaces" and does not depend on physical drawing in graphs.

The appearance of time variables related to spacial coordinates in equations does not mean that time and space are equivalent and that time no longer means time as we initially conceptualize it. To believe otherwise is to treat equations as floating abstractions cut off from the facts that give rise to them, stealing the concept of time by denying the concepts on which the physical theories depend. The objectivity of the concept of space-time is the same as any other valid higher level abstract concept -- the concept is objectively based on lower level abstractions in a hierarchy that is ultimately traced back to sense perception of entities in reality, in this case through complex procedures and validations of concepts and principles of science.